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Lesson 1. Conventional productivity and

efficiency concepts

Diego Prior

Dpt. of Business

Productive efficiency and innovation

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Objectives for the Subject

1. Examine the conceptual framework that

underpins productivity measurement

2. Introduce the principal methods

• Index Numbers

• Non-parametric methods

• Parametric methods

• Examine these techniques, relative merits,

necessary assumptions and guidelines for

their applications

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Objectives for the Subject

3. Work with computer programs

4. Briefly review some case studies and real

life applications

5. Briefly review some advanced topics

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Main Reference:

An Introduction to Efficiency and

Productivity Analysis (2nd Ed.)

Coelli, Rao, O’Donnell and Battese

Springer, 2005

Supplemented with material from other

published articles and books

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Outline for today

• 1.1. Importance of productivity measures.

• 1.2. Conventional concepts of productivity,

technical efficiency and technical change.

• 1.2.1. Non-frontier methodologies.

• 1.2.2. Frontier methodologies.

• 1.3. Effectiveness, quality and efficiency.

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1.1. Importance of productivity

measures.• Performance measurement

– Productivity measures

• Mainly using partial productivity measures

• Cost, revenue and profit ratios

• Performance of public services and utilities

• Aggregate Level

– Growth in per capita income

– Labour and total factor productivity growth

– Sectoral performance

• Labour productivity

• Share in the total economy

• Industry Level

– Performance of firms

– Market and non-market goods and services

– Efficiency and productivity

– Banks, credit unions, manufacturing firms, agricultural farms, schools and universities, hospitals, aged care facilities, etc.

• Need to use appropriate methodology to benchmark performance

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Productivity:

• It measures the level of output per unit of input.

• Partial productivity measures – output per person employed;

output per hour worked; output per hectare etc.

• Total factor productivity measures – Productivity measure which

involves all the factors of production

Efficiency:

(i) How much more can we produce with a given level of inputs?

(ii) How much input reduction is possible to produce a given level of

observed output?

(iii) How much more revenue can be generated with a given level of

inputs? Similarly how much reduction in input costs be achieved?

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Partial indicators

• Can be misleading

• Consider two clothing factories (A and B)

• Labour productivity could be higher in firm

A – but what about use of capital and

energy and materials?

• Unit costs could be lower in firm B – but

what if there exists significant differences

in terms of quality?

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1.2. Frontier and non frontier methods

• The terms productivity and efficiency are

different:

• Productivity = output/input

• Efficiency generally relates to some form

benchmark or target

• An example – where for firm B productivity

rises but efficiency falls:

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Productivity?

• productivity = output/input

• What to do if we have more than one input

and/or output?

– partial productivity measures

– aggregation

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Example

• Two firms producing t-shirts using labour and capital (machines).

• The partial productivity ratios conflict.

firm labour (x1)

capital (x2)

output (q)

q/x1 q/x2

A 2 2 200 100 100

B 4 1 200 50 200

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Non-frontier total factor productivity (TFP)

• Use an aggregate measure of input:

TFP = y/(a1x1+a2x2)

• What should we use as the weights? – prices?

• Data: Labour wage = $80 per day and

Rental price of the machines = $100 per day

• Calculation:

TFPA = 200/(80×2+100×2) = 200/360 = 0.56

TFPB = 200/(80×4+100×1) = 200/420 = 0.48

=>A is more productive using this measure.

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Example of output oriented efficiency analysis

x q1 q2 q1/x q2/x

A 10 40 2 4 0.2

B 10 20 5 2 0.5

C 10 10 20 1 2

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Example of frontier analysis

(non-convex technology):

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BENCHMARK ANALISIS (TWO INPUTS, ONE OUTPUT)

FIRM

OUTPUT

(y)

x 1/y

x 2/y

Quality level

A 1 4 1 2 (good)

B 1 2 2 1 (bad)

C 1 1 3 2 (good)

D 1 4 5 2 (good)

E 1 5 3 2 (good)

F 1 2 4 2 (good)

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Production Technology

• We assume that there is a production technology that allows transformation of a vector of inputs into a vector of outputs

S = {(x,q): x can produce q}.

• Technology set is assumed to satisfy some basic axioms.

• It can be equivalently represented by– Output sets

– Input sets

– Output and input distance functions

• A production function provides a relationship between the maximum feasible output (in the single output case) for a given set of input

• Single output/single input; single output/multiple inputs; multi-output/multi-input

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Output and Input sets

• Output set P(x) for a given vector of inputs, x, is the set of all possible output vectors q that can be produced by x.

P(x) = {q: x can produce q} = {q : (x,q) ∈ S}

– P(x) satisfies a number of intuitive properties including: nothing can be produced from x; set is closed, bounded and convex

– Boundary of P(x) is the production possibility curve

• An Input set L(q) can be similarly defined as set of all input vectors x that can produce q.

L(q) = {x: x can produce q} = {x: (x , q) ∈ S}

– L(q) satisfies a number of important properties that include: closed and convex

– Boundary of L(q) is the isoquant curve

• These sets are used in defining the input and output distance

functions

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Frontier concepts: Output

Distance Function

Do(x,y)

The value of the distance function is

equal to the ratio δ=0A/0B.

y1A

y2A

B

CA

y10

y2

P(x)

•

•• PPC-P(x)

Technical Efficiency Measure:

TE = 0A/0B = do(x,q)

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Frontier concepts: Input

Distance Function

Di(x,y)

The value of the distance function is

equal to the ratio ρ=0A/0B.

Isoq-L(y)

x1A

x2A

B

C

A

x1 0

x2

L(y) •

•

•

Technical Efficiency measure:

= TE = 1/di(x,q) = OB/OA

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1.3. Effectiveness and efficiency

• The production technology defines the

technological constraints.

• The objective of the firm could be to maximise

profit

• Or minimise costs when outputs are fixed

• Or maximise revenue when inputs are fixed

• Or … other strategical goals (i.e. to maximize

the market share or to increase the customers’

loyalty)

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Profit maximisation

frontier

q

x

Profit max

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Cost minimisation

• The firm must produce output, q0

• Minimum cost is defined as:

( , ) min { : ( , ) }c S′= ∈x

q w w x x q

Isoquant (q=q0)

x2

x1

Cost min

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Revenue maximisation

• The firm has input allocation, x0

• Maximum revenue is defined as:

( , ) max { : ( , ) }r S′= ∈q

p x p q x q

Revenue max

PPC (x=x0)

y2

y1

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Returns to Scale

• A production technology exhibits constant returns

to scale (CRS) if a Z% increase in inputs results in

Z% increase in outputs (ε = 1).

• A production technology exhibits increasing

returns to scale (IRS) if a Z% increase in inputs

results in a more than Z% increase in outputs (ε >

1).

• A production technology exhibits decreasing

returns to scale (DRS) if a Z% increase in inputs

results in a less than Z% increase in outputs (ε <

1).

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Returns to scale

q

x

DRS

IRS

CRS

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Efficiency Measures

• Using the distance functions defined so far, we can define:

– Technical efficiency

– Allocative efficiency

– Economic efficiency

• A firm is said to be technically efficient if it operates on the frontier of the production technology

• A firm is said to be allocatively efficient if it makes efficient allocation in terms of choosing optimal input and output combinations.

• A firm is said to be economically efficient if it is both technically and allocatively efficient.

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Example of Technical Efficiency

Output orientation: TEO=DA/DB

Input orientation: TEI=EC/EA

q

x

Frontier

A•

•

•

B

CE

D

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Non-frontier total factor productivity (TFP)

• Can decompose TFP difference between 2

firms (at one point in time) into 3 types of

efficiency:

– technical efficiency,

– allocative efficiency and

– scale efficiency.

• Yes, we can! but …

“we need to know the technology”

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How do we measure efficiency?

• Depends upon the type of data available for the measurement purpose.

• Three types:

– Observed input and output data for a given firm over two periods or data for a few firms at a given point of time (time-series data);

– Observed input and output data for a large sample of firms from a given industry (cross-sectional data)

– Panel data on a cross-section of firms over time

• In the first case measurement is limited to productivity measurement based on restrictive assumptions.

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Overview of Methods:

• Non-frontier index numbers (IN)

– Price and quantity index numbers used in

aggregation.

• Frontier non-parametric methods (convex

or non-convex)

• Frontier parametric methods

(deterministic or stochastic methods (SFA)

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Relative merits of non-frontier index numbers:

• Advantages:

– only need 2 observations

– transparent and reproducible

– easy to calculate

• Disadvantages:

– need price information

– Lack of theoretical framework

– cannot decompose in substantive terms

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Relative merits of frontier methods:

• Advantages of non-parametric methods:

− no need to specify functional form or distributional forms for errors

− easy to accommodate multiple outputs

• Advantages of parametric methods:

− attempts to account for data noise

− can conduct hypothesis tests with statisticalsignificance

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And that’s all folks!